Let's break down the given function $f(x) = 3[-2(x + 4)]^2 - 1$ and answer the questions on the chart.

1. Parent Graph:

   • The parent function is $f(x) = x^2$, which is a standard quadratic function (a parabola).

2. Vertical Translation:

   • The function has a vertical shift of $-1$, which means the graph moves down by 1 unit.

3. Horizontal Translation:

   • Inside the function, we have $(x + 4)$. This represents a horizontal shift to the left by 4 units.

4. Stretch/Compress (Vertical):

   • The coefficient of 3 outside the squared term indicates a vertical stretch by a factor of 3.

5. Stretch/Compress (Horizontal):

   • The factor $-2$ inside the squared term represents a horizontal compression by a factor of $\frac{1}{2}$ (since $|2|$ is the horizontal compression factor).

6. Flipped?

   • The negative sign inside the squared term indicates a reflection across the y-axis (upside down along the horizontal axis).

Summary for the chart:

• Vertical Translation: The graph moved down 1 unit.
• Horizontal Translation: The graph moved left 4 units.
• Vertical Stretch: The graph is stretched vertically by a factor of 3.
• Horizontal Compression: The graph is compressed horizontally by a factor of 2.
• Flipped: The graph is reflected (flipped upside down).

Would you like more details or clarification on any of the transformations?

Here are 5 related questions:

1. How would the graph change if the coefficient outside was negative, like $-3$?
2. What would happen if the horizontal shift was $(x - 4)$ instead of $(x + 4)$?
3. How do you identify vertical versus horizontal stretches or compressions in a function?
4. How can you determine the direction of a graph’s flip based on its equation?
5. What does the general form $a(x - h)^2 + k$ tell you about transformations?

Tip: Always check both inside and outside the function to identify horizontal and vertical transformations separately.